sparse grid method in the libor market model. option valuation and the

There is an important difference between Itô’s formula and its deterministic counterpart.
The laws of stochastic calculus 8 stipulate that dWt
2 = σt 2 dt, where W t is a
Brownian motion. As a result, the second derivative term with respect to a stochastic
factor is of the same order of accuracy as ∂f(t,X t)
∂t
and cannot be neglected.
As a demostration of Itô’s lemma, let’s try to acquire the process for geometric
Brownian motion as given by X t = W t and f(t, W t ) = S t = S 0 exp ( )
νt + σW t ,
which is also (2.11). The partial derivatives of S t with respect to time and Brownian
motion are:
and therefore:
∂f(t, X t )
∂t
∂f(t, X t )
∂ 2 f(t, X t )
= νS t = σS t
∂X t ∂Xt
2 = σ 2 S t dt
dS t = (ν + 1 2 σ2 )S t dt + σS t dW
}{{} t
} {{ }
σ(t,S t )
µ(t,S t)
Thus, the process for prices given by geometric Brownian motion is an Itô process. We
shall call ν + 1 2 σ2 - an instantaneous expected return and ν - an overall expected return.
Black-Scholes-Merton Model Having accepted a log-normal distribution for the
continuous process of an underlying asset, we have eventually prepared ourselves for
discussion of Black-Scholes-Merton approach to option valuation. In their seminal paper
[Black and Scholes, 1973], Black and Scholes have suggested a model that dwells
on several assumptions:
• Trading proceeds continuously;
• Stock prices follow a log-normal distribution, the process described by (2.15);
• Absense of arbitrage opportunities;
• No rebalancing costs;
• Limitless lending and borrowing is allowed;
Having imposed this setting, Black and Scholes argued that a short position in a derivative
c t can be made riskless in case it is a part of a portfolio Π t such that:
Π t = −c t + ∆ t S t (2.17)
Both c t and S t depend on the same source of uncertainty, which is a Brownian Motion
term W t of the stock price process. Therefore if Π t is constructed by taking reverse positions
in a derivative and a certain amount ∆ t of its underlying, the Brownian motion
terms cancel, eliminating the risk. Π t is called a Hedge portfolio. According to our
no-arbitrage assumption, a position in a porfolio that has no risk must be of the same
nature as a position in a risk-free cash bond B t , i.e. earn a risk-free interest rate during
its lifetime:
dΠ t = rΠ t dt
8 Namely, the quadric variation result. See [Etherage, 2002].
11